1. Field of the Invention
The present invention relates to measurement of normalized Fourier coefficients for analyzing a waveform when measuring exposure in real time using an integrating photometric detector with respect to an intensity of an optical signal that varies periodically with time.
2. Description of Related Art
When measuring in real time an intensity of an optical signal that varies periodically with time (t) using an integrating photometric detector, Fourier coefficients are used to analyze the waveform. Assuming that there is no error in such measuring device, an intensity I′(t) of light measured as an electric signal such as voltage or current by the photometric detector with respect to a specific wavelength can be expressed by the following equation consisting of a mean value I0′ of the light intensity, normalized Fourier coefficients α′2n and β′2n, and a period (T):
                                          I            ′                    ⁡                      (            t            )                          =                                            I              ′                        0                    ⁢                                    {                              1                +                                                      ∑                                          n                      =                      1                                        N                                    ⁢                                      [                                                                                            α                                                      2                            ⁢                            n                                                    ′                                                ⁢                                                  cos                          ⁡                                                      (                                                          4                              ⁢                              π                              ⁢                                                                                                                          ⁢                              n                              ⁢                                                                                                                          ⁢                                                              t                                /                                T                                                                                      )                                                                                              +                                                                        β                                                      2                            ⁢                            n                                                    ′                                                ⁢                                                  sin                          ⁡                                                      (                                                          4                              ⁢                              π                              ⁢                                                                                                                          ⁢                              n                              ⁢                                                                                                                          ⁢                                                              t                                /                                T                                                                                      )                                                                                                                ]                                                              }                        .                                              (        1        )            Where, 2N is a natural number that indicates a maximum degree of the normalized Fourier coefficients other than 0.
A representative example expressed by the equation (1) is a value of intensity of light measured with a photometric detector in a rotating optical element ellipsometer. The rotating optical element ellipsometer measures an intensity of light with a photometric detector in a state that an optical element such as a linear polarizer or a compensator rotates at a constant speed. In particular, an ellipsometer, a non-destructive and non-contacting realtime measuring device, is widely used to evaluate physical properties of nanofilms fabricated in various nanofilm processes such as a semiconductor device and a flat panel display.
In the case of rotating polarizer and rotating analyzer ellipsometers of various types of the rotating optical element ellipsometer, they have the same main elements: a light source, a polarizer (linear polarizer), a sample, an analyzer (linear polarizer), and a photometric detector, but measurement is carried out while only one of the polarizer and the analyzer rotates at a constant speed. This corresponds to a case that N is 1 in the equation (1), the intensity of the light measured by the photometric detector, since all other than the normalized Fourier coefficients of quadratic terms such as α′2 and β′2 have the value of 0. In a case of a single rotating compensator ellipsometer, a compensator is added between the polarizer and the sample or between the sample and the analyzer of the aforementioned measuring device and this compensator alone rotates at a constant speed to carry out the measurement. This corresponds to a case that N is 2 in the equation (1), the intensity of the light measured by the photometric detector, since only the Fourier coefficients of quadratic and biquadratic terms such as α′2, β′2, α′4 and β′4 are not 0. In a case of a dual rotating compensator ellipsometer, the device consists of a light source, a polarizer, a compensator, a sample, a compensator, an analyzer and a photometric detector, and two compensators rotate at a relatively constant speed to carry out the measurement. In this case, N is 16 since the Fourier coefficients of effective terms of maximum degree in equation (1) are α′32 and β′32.
In the ellipsometers, it is very important to obtain, more correctly, normalized Fourier coefficients α′2n and β′2n from the waveform of the light intensity measured by the photometric detector like the equation (1). The realtime rotating optical element ellipsometer most widely used in recent generally employs a CCD detector array or a photodiode detector array as the photometric detector. These photometric detectors are called as an integrating photometric detector since they are in proportion to not only the light intensity but also an integration time Tint. This integrating photometric detector may carry out the measurement in a proper condition by properly reducing or increasing the integration time when the light amount is too much or insufficient upon the measurement, but the integration time upon the measurement should always be set to equal or larger than a minimum integration time of the relevant photometric detector. A value of the light amount, i.e. an exposure Sj, measured under a condition of measuring the light intensity, which varies periodically with time, M times with a constant interval T/M during the period T by the equation (1) using the aforementioned integrating photometric detector and matching the integration time correctly to the time interval, i.e. under a specific condition that Tint=T/M, is expressed as follows:
                                                                                          S                  j                                =                                ⁢                                                      ∫                                                                  (                                                  j                          -                          1                                                )                                            ⁢                                              T                        /                        M                                                                                    jT                      /                      M                                                        ⁢                                                                                    I                        ′                                            ⁡                                              (                        t                        )                                                              ⁢                                                                                  ⁢                                          ⅆ                      t                                                                                  ;                              (                                                      j                    =                    1                                    ,                  2                  ,                  3                  ,                  …                  ⁢                                                                          ,                  M                                )                                                                                        =                            ⁢                                                                                                                  I                        ′                                            0                                        ⁢                    T                                    M                                +                                                      ∑                                          n                      =                      1                                        N                                    ⁢                                                                                                                                          I                            ′                                                    0                                                ⁢                        T                                                                    2                        ⁢                        n                        ⁢                                                                                                  ⁢                        π                                                              ⁢                                          sin                      ⁡                                              (                                                                              2                            ⁢                            n                            ⁢                                                                                                                  ⁢                            π                                                    M                                                )                                                                                                                                                                                  ⁢                                                {                                                                                    α                                                  2                          ⁢                          n                                                ′                                            ⁢                                              cos                        ⁡                                                  [                                                                                    2                              ⁢                              n                              ⁢                                                                                                                          ⁢                                                              π                                ⁡                                                                  (                                                                                                            2                                      ⁢                                      j                                                                        -                                    1                                                                    )                                                                                                                      M                                                    ]                                                                                      +                                                                  β                                                  2                          ⁢                          n                                                ′                                            ⁢                                              sin                        ⁡                                                  [                                                                                    2                              ⁢                              n                              ⁢                                                                                                                          ⁢                                                              π                                ⁡                                                                  (                                                                                                            2                                      ⁢                                      j                                                                        -                                    1                                                                    )                                                                                                                      M                                                    ]                                                                                                      }                                .                                                                        (        2        )            By solving simultaneous equations like the equation (2) with respect to the normalized Fourier coefficients, a formula of the normalized Fourier coefficients α′2n and β′2n expressed by the exposure Sj is obtained and this is called as the Hadamard transform. The equation (2) has been used as a representative method to be able to obtain the Fourier coefficients of the equation (1) in a case of using the integrating photometric detector in the rotating optical element ellipsometer. However, since in an actual integrating photometric detector, the photometric detector does not response during the time of reading out the light amount accumulated in each pixel during the integration time and initializing the status, i.e. read time Tr, the exposure of the equation (2) was corrected, in consideration of this fact, as follows:
                                                        S              j                        =                                          ∫                                                                            (                                              j                        -                        1                                            )                                        ⁢                                          T                      /                      M                                                        +                                      T                    r                                                                    jT                  /                  M                                            ⁢                                                                    I                    ′                                    ⁡                                      (                    t                    )                                                  ⁢                                                                  ⁢                                  ⅆ                  t                                                              ;                                          ⁢                      (                                          j                =                1                            ,              2              ,              3              ,              …              ⁢                                                          ,              M                        )                          ,                            (        3        )            and an equation obtained by the first-order approximation to Tr is used on the assumption that the read time Tr is very shorter than the measuring time interval T/M.
In the case of a conventional rotating polarizer or analyzer ellipsometer using the Hadamard transform, in the equation (1), T is a period of mechanical turn of a polarizer or analyzer and N is 1 as already described above. Also, although a minimum value of the measurement number M during the period T is 6, the measurement number M is increased to 8 since it is required to additionally measure β′4 so as to found whether the system is normal or not. At this time, since the values of the exposure measured at each section are symmetric with respect to T/2 period, four unknown coefficients I′0, α′2, β′2 and β′4 can be obtained from four simultaneous equations consisting of only S1, S2, S3 and S4 measured in the first half.
Meanwhile, in the case of a conventional single rotating compensator ellipsometer using the Hadamard transform, T is a period of mechanical turn of a compensator and N is 2. Also, although a minimum value of the measurement number M during the period T is 10, the measurement number M is increased to 16 since it is required to additionally measure β′8 so as to found whether the system is normal or not. Like the previous case, a value of the simultaneous equation was used to obtain six unknown coefficients I′0, α′2, β′2, α′4, β′4 and β′8 from the measured values of Sj (j=1, 2, 3, . . . , 8) in consideration of the symmetry of the measured values of the exposure.
Finally, in the case of a conventional dual rotating compensator ellipsometer using the Hadamard transform, 36 unknown coefficients were obtained in a very complex form by solving 36 simultaneous equations.
In the conventional rotating optical element ellipsometer using the Hadamard transform, the maximum degree 2N of the Fourier coefficients other than 0 is determined different according to the kind of the ellipsometer, there was used a complex equation for obtaining mean value I′0 of the light intensity and normalized Fourier coefficients α′2n and β′2n from the value Sj of the exposure measurement by solving the simultaneous equation obtained by substituting the equation (1) into equation (2), and finally an error of the read time was corrected using the first-order approximation equation with respect to read time obtained from the equation (3).
With development of manufacturing technique of the integrating photometric detector, the read time and the minimum integration time have been reduced to 1 ms or less and a measurement sensitivity has been notably improved. Since it is possible to reduce the measurement time to several ms for the faster measurement, ratio of the read time to the measurement time interval is gradually increased. Therefore, a correction method more accurate than the conventional primary approximation to the read time is required for accurate measurement. Also, when using the conventional Hadamard transform, since sum of the read time and the integration time should be set to exactly accord with the measurement time interval, the light amount easily reaches to a saturation state within the short integration time when the light intensity is too high and thus some of the light beam should be shielded by using additional optical elements such as an iris diaphragm and a neutral density filter (ND filter).